Multi-valued Functions in Computability Theory

Abstract

Multi-valued functions are common in computable analysis (built upon the Type 2 Theory of Effectivity), and have made an appearance in complexity theory under the moniker search problems leading to complexity classes such as \(\textrm{PPAD}\) and \(\textrm{PLS}\) being studied. However, a systematic investigation of the resulting degree structures has only been initiated in the former situation so far (the Weihrauch-degrees).

A more general understanding is possible, if the category-theoretic properties of multi-valued functions are taken into account. In the present paper, the category-theoretic framework is established, and it is demonstrated that many-one degrees of multi-valued functions form a distributive lattice under very general conditions, regardless of the actual reducibility notions used (e.g., Cook, Karp, Weihrauch).

Beyond this, an abundance of open questions arises. Some classic results for reductions between functions carry over to multi-valued functions, but others do not. The basic theme here again depends on category-theoretic differences between functions and multi-valued functions.